The PI plans to investigate projects in four areas, related by the theme of nonlinear PDE's and their interaction and applications to complex and symplectic geometry. In the first project, the PI will work on Donaldson's program of extending Yau's theorem in Kahler geometry to symplectic 4-manifolds. The ultimate goal is to tackle some fundamental questions in symplectic and almost complex geometry, such as: given an almost complex 4-manifold, when does there exist a compatible symplectic form? The second project deals with the constant scalar curvature equation on Kahler manifolds. It is believed that the existence of a solution to this nonlinear PDE should be equivalent to the algebraic stability of the manifold. The PI will examine this via the natural energy functionals associated to this problem, building on his work with Phong, Song and Sturm. The third project concerns the Kahler-Ricci and Calabi flows and aims to elucidate the relationship between the convergence properties of these parabolic flows and conditions of stability and positive curvature. The final project also involves parabolic PDE's: the geometric flows of Donaldson which arise naturally from considerations of moment maps and diffeomorphism groups. The PI will investigate the behavior of these flows, on Kahler and hyperkahler manifolds, and possible applications to the study of canonical metrics and the space of symplectic forms.
The fundamental laws of physics are described, in the language of mathematics, by differential equations. Understanding the behavior of solutions to differential equations (such as Einstein's equations) is key to unravelling the mystery of the geometry and structure of the universe. This proposal concerns equations which arise naturally in the study of geometry and are related to, and inspired by, the physical laws. There is a complex interaction between such differential equations, which are described locally, and the underlying geometry, which is global. This project aims to investigate this rich interplay between local and global and in the process touch on some basic problems in mathematics: what are the natural structures in geometry, and how do we find and describe them?
